Optimal. Leaf size=77 \[ \frac{2 \sqrt{e (c+d x)} \left (a+b \sinh ^{-1}(c+d x)\right )^4}{d e}-\frac{8 b \text{Unintegrable}\left (\frac{\sqrt{e (c+d x)} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{\sqrt{(c+d x)^2+1}},x\right )}{e} \]
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Rubi [A] time = 0.181656, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\left (a+b \sinh ^{-1}(c+d x)\right )^4}{\sqrt{c e+d e x}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\left (a+b \sinh ^{-1}(c+d x)\right )^4}{\sqrt{c e+d e x}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \sinh ^{-1}(x)\right )^4}{\sqrt{e x}} \, dx,x,c+d x\right )}{d}\\ &=\frac{2 \sqrt{e (c+d x)} \left (a+b \sinh ^{-1}(c+d x)\right )^4}{d e}-\frac{(8 b) \operatorname{Subst}\left (\int \frac{\sqrt{e x} \left (a+b \sinh ^{-1}(x)\right )^3}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{d e}\\ \end{align*}
Mathematica [A] time = 8.6694, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \sinh ^{-1}(c+d x)\right )^4}{\sqrt{c e+d e x}} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.199, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) ^{4}{\frac{1}{\sqrt{dex+ce}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{4} \operatorname{arsinh}\left (d x + c\right )^{4} + 4 \, a b^{3} \operatorname{arsinh}\left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \operatorname{arsinh}\left (d x + c\right )^{2} + 4 \, a^{3} b \operatorname{arsinh}\left (d x + c\right ) + a^{4}}{\sqrt{d e x + c e}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{asinh}{\left (c + d x \right )}\right )^{4}}{\sqrt{e \left (c + d x\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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